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Jun 29, 2018
06/18

by
I. Krasovsky

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We consider the spectrum of the almost Mathieu operator $H_\alpha$ with frequency $\alpha$ and in the case of the critical coupling. Let an irrational $\alpha$ be such that $|\alpha-p_n/q_n|0$.

Topics: Spectral Theory, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1602.08624

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4.0

Jun 30, 2018
06/18

by
T. Claeys; I. Krasovsky

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We study asymptotic behavior for determinants of $n\times n$ Toeplitz matrices corresponding to symbols with two Fisher-Hartwig singularities at the distance $2t\ge0$ from each other on the unit circle. We obtain large $n$ asymptotics which are uniform for $0

Topics: Complex Variables, Mathematics, Mathematical Physics, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1403.3639

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36

Sep 18, 2013
09/13

by
P. Deift; A. Its; I. Krasovsky

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We obtain asymptotics for Toeplitz, Hankel, and Toeplitz+Hankel determinants whose symbols possess Fisher-Hartwig singularities. Details of the proofs will be presented in another publication.

Source: http://arxiv.org/abs/0809.2420v2

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47

Jul 20, 2013
07/13

by
P. Deift; A. Its; I. Krasovsky

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The authors use Riemann-Hilbert methods to compute the constant that arises in the asymptotic behavior of the Airy-kernel determinant of random matrix theory.

Source: http://arxiv.org/abs/math/0609451v2

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60

Sep 21, 2013
09/13

by
P. Deift; A. Its; I. Krasovsky

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We study the asymptotics in n for n-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general non-degenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.

Source: http://arxiv.org/abs/0905.0443v3

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57

Sep 22, 2013
09/13

by
T. Claeys; A. Its; I. Krasovsky

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We obtain asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter $t$. For $t$ positive, the symbols are regular so that the determinants obey Szeg\H{o}'s strong limit theorem. If $t=0$, the symbol possesses a Fisher-Hartwig singularity. Letting $t\to 0$ we analyze the emergence of a Fisher-Hartwig singularity and a transition between the two different types of asymptotic behavior for Toeplitz determinants. This transition is described by a...

Source: http://arxiv.org/abs/1004.3696v2

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102

Jul 19, 2013
07/13

by
P. Deift; I. Krasovsky; J. Vasilevska

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We obtain "large gap" asymptotics for a Fredholm determinant with a confluent hypergeometric kernel. We also obtain asymptotics for determinants with two types of Bessel kernels which appeared in random matrix theory.

Source: http://arxiv.org/abs/1005.4226v3

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Sep 23, 2013
09/13

by
P. Deift; A. Its; I. Krasovsky

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The authors analyze the asymptotics of eigenvalues of Toeplitz matrices with certain continuous and discontinuous symbols. In particular, the authors prove a conjecture of Levitin and Shargorodsky on the near-periodicity of Toeplitz eigenvalues.

Source: http://arxiv.org/abs/1110.4089v2

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74

Sep 21, 2013
09/13

by
A. Its; I. Krasovsky

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We obtain asymptotics in n for the n-dimensional Hankel determinant whose symbol is the Gaussian multiplied by a step-like function. We use Riemann-Hilbert analysis of the related system of orthogonal polynomials to obtain our results.

Source: http://arxiv.org/abs/0706.3192v3

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92

Jul 20, 2013
07/13

by
I. Krasovsky

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We review the asymptotic behavior of a class of Toeplitz (as well as related Hankel and Toeplitz + Hankel) determinants which arise in integrable models and other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and how a transition between them is related to the Painleve V equation. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits, to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and...

Source: http://arxiv.org/abs/1007.1128v3

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94

Sep 20, 2013
09/13

by
P. Deift; A. Its; I. Krasovsky

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We review some history and some recent results concerning Toeplitz determinants and their applications. We discuss, in particular, the crucial role of the two-dimensional Ising model in stimulating the development of the theory of Toeplitz determinants.

Source: http://arxiv.org/abs/1207.4990v3

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Sep 22, 2013
09/13

by
T. Claeys; A. Its; I. Krasovsky

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We study Fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher order analogues of the Airy kernel and are built out of functions associated with the Painlev\'e I hierarchy. The Fredholm determinants related to those kernels are higher order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a...

Source: http://arxiv.org/abs/0901.2473v1

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Sep 20, 2013
09/13

by
P. Deift; A. Its; I. Krasovsky; X. Zhou

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In this paper we consider an asymptotic question in the theory of the Gaussian Unitary Ensemble of random matrices. In the bulk scaling limit, the probability that there are no eigenvalues in the interval (0,2s) is given by P_s=det(I-K_s), where K_s is the trace-class operator with kernel K_s(x,y)={sin(x-y)}/{\pi(x-y)} acting on L^2(0,2s). We are interested particularly in the behavior of P_s as s tends to infinity...

Source: http://arxiv.org/abs/math/0601535v1

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98

Jul 20, 2013
07/13

by
I. Krasovsky

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We outline an approach recently used to prove formulae for the multiplicative constants in the asymptotics for the sine-kernel and Airy-kernel determinants appearing in random matrix theory and related areas.

Source: http://arxiv.org/abs/1007.1135v1